let  the end points of chord be (x1,y1) and (x2,y2).
mid point: (x,y)= ((x1+x2)/2),(y1+y2)/2)

centre= (3/2,-1) and radius is 1/2.
Since, the angle subended at the centre is of 90.
So, distance between the end-points of chord will be sqrt(2). (check diagramatically).
so, (x1-x2)^2 + (y1-y2)^2 = 1/2.
put the value of x1= 2x-x2 and similarly for y1.
so, we get (x-x2)^2 + (y-y2)^2 = 1/8.

now, join centre and mid point, and see carefully that angles in each half are two angles 45 and 1 angle is of 90.
apply, pythogoras theorem in half section.
((x-3)^2 + (y+2)^2)  + ((x-x2)^2 + (y-y2)^2) =1/4.

solve both equations:
u'll get x^2 -6x +y^2 +4x +13 =1/8.

If still you have any doubt do tell me...